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Astron. Astrophys. 318, 947-956 (1997)

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Appendix A: complete expressions of the ions and neutrals velocities

In Hénoux and Somov (1991), velocities of electrons, ions and neutrals have been computed, for an axially symmetrical magnetic field, in the three fluids approximation, by solving the equations that express the balance of the horizontal forces acting per unit volume on each particle fluid, i.e.

[EQUATION]

Here [FORMULA] is the friction force due to particles l acting on a particule k, and [FORMULA]. [FORMULA] and [FORMULA] are the magnetic and electric field vectors. [FORMULA] and [FORMULA] are the drift velocity, density and mass of a particle k. [FORMULA] is the coefficient of friction between particles k and l. [FORMULA] is the velocity field in the convective zone.

The horizontal velocities of ions and neutrals derived from the equations above are (Hénoux and Somov, 1991):

[EQUATION]

The ions and neutrals velocities in these equations are relatives to the velocities in the convective zone, since all components of the vector [FORMULA] have been replaced by [FORMULA]. The function [FORMULA] is given by:

[EQUATION]

where [FORMULA] is the electric conductivity parallel to [FORMULA]. [FORMULA] [FORMULA] where [FORMULA] and [FORMULA] are the collision frequencies between electrons, ions and neutrals and between electrons and ions. [FORMULA] and [FORMULA] are the contributions to [FORMULA] and [FORMULA] of the inertial terms. They are given by:

[EQUATION]

The summation is made over all species, electrons, ions and neutrals. The azimuthal and radial current densities are found to be

[EQUATION]

where [FORMULA] is the vertical current density. [FORMULA] is defined by [FORMULA] where [FORMULA] is defined by :

[EQUATION]

The subscripts [FORMULA] refer to the azimuthal, radial and vertical components of the magnetic and velocity fields. The subscripts i, e and n correspond respectively to the ions, electrons, and neutrals.

The equation (A.11) can be rewritten as

[EQUATION]

Appendix B: radial velocities of ions and neutrals

Assuming no radial motions in the convective zone and that the radial component of [FORMULA] is null, we can write the radial velocities of neutrals and ions as :

[EQUATION]

Neglecting the contribution of electrons and in the hypothesis that [FORMULA], we obtain from equations (A.7) and (A.12) [FORMULA]. Consequently the radial velocities of ions and neutrals are given by

[EQUATION]

Appendix C: azimuthal velocity of neutrals

The radial dependence of [FORMULA] can be derived from equations (A.5) and (A.10). Considering successively the various terms on the RHS of eq.(A.5) we show below that the last term is the dominant term on the RHS of this equation. i) The third term, that is indeed null for a vertical magnetic field, gives the effect, on the azimuthal velocity of neutrals, of the collisions with charges, ions and electrons, moving along the lines of force. Since the velocities that neutrals achieve under the effect of these collisions are also parallel to the lines of force, they do not imply any electromagnetic force and the third term can be omitted. Ignoring this term implies that the azimuthal velocity in the LHS of the simplified resulting equation is the azimuthal component of the velocity vector component that is perpendicular to the magnetic field vector. ii) The second term gives the limit of the azimuthal velocity of neutrals, that does result from the collisions with the charged particles that carry the azimuthal current [FORMULA]. In the region of generation of the radial currents. This velocity is orders of magnitude smaller that the azimuthal velocity of neutrals and can therefore be neglected. iii) In the thin flux-tube approximation the first term on the RHS is null. Then the azimuthal velocity of neutrals can be rewritten as

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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